Download Limitations on the superposition principle: superselection

Eur. J. Phys. 19 (1998) 237–243. Printed in the UK
PII: S0143-0807(98)87270-X
Limitations on the superposition
principle: superselection rules in
non-relativistic quantum mechanics
C Cisneros†, R P Martı́nez-y-Romero‡,
H N Núñez-Yépez§k, and A L Salas-Brito¶
† Laboratorio de Cuernavaca, Instituto de Fı́sica, Universidad Nacional Autónoma de
México, Cuernavaca Mor, Mexico
‡ Departamento Fı́sica, Facultad de Ciencias, Universidad Nacional Autónoma de
México, Apartado Postal 21-726, Coyoacán 04000 DF, Mexico
§ Instituto de Fı́sica ‘Luis Rivera Terrazas’, Benemérita Universidad Autónoma de
Puebla, Apartado Postal J-48, CP 72570, Puebla Pue, Mexico
Received 12 September 1997, in final form 16 February 1998
Abstract. The superposition principle is a very basic ingredient of quantum theory. What may
come as a surprise to many students, and even to some practitioners of the quantum craft, is
that superposition has limitations imposed by certain requirements of the theory. The discussion
of such limitations arising from the so-called superselection rules is the main purpose of this
paper. Some of their principal consequences are also discussed. The univalence, mass and
particle number superselection rules of non-relativistic quantum mechanics are also derived
using rather simple methods.
1. Introduction
Linearity is one of the most basic properties of quantum theory: any set
Pof states, |αi, of a
quantum system may be linearly superposed to obtain a new state |Ai = α aα |αi, where the
aα are complex numbers, the probability amplitudes for the system to be found in the state |αi.
Linear combinations like |Ai are usually referred to as coherent superpositions or pure states
(Sakurai 1985). A brief discussion of coherent as contrasted to incoherent superpositions or
mixtures, is given in section 2.
However, despite being fundamental, it is easy to realize that the superposition principle
cannot hold unrestrictedly in every situation. For example, think of photons and electrons:
then, no matter what we have just said about the superposition principle, no one has ever
made sense of any superposition of electronic with photonic states (Sudbery 1986, SalasBrito 1990). How can we understand this peculiarity within the general structure of quantum
theory? The justification for such fact was found long ago (Wick et al 1952) and involves
k On sabbatical leave from: Departamento de Fı́sica, UAM-Iztapalapa, Mexico; e-mail: [email protected] .
¶ Corresponding author. On sabbatical leave from: Lab. de Sistemas Dinámicos, Departamento de Ciencias
Básicas, UAM-Azcapotzalco, Mexico; e-mail: [email protected] or [email protected] .
0143-0807/98/030237+07$19.50
© 1998 IOP Publishing Ltd
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the curious phenomenon referred to as a superselection rule. To give a brief description,
a superselection rule exists whenever there are limitations to the superposition principle
(Kaempfer 1965, Sudbery 1986), that is, whenever certain superpositions cannot be physically
realizable. Moreover, the very existence of superselection rules explains why we can treat
certain observables like the mass of a particle as parameters rather than as full-fledged operators
in non-relativistic quantum mechanics (NRQ).
It is somewhat peculiar that despite the additional insight into the subtleties of quantum
theory that might be offered by superselection rules and of the fundamental role they have
played in explaining puzzling phenomena in various areas of physics (Streater and Wightman
1964, Pfeifer 1980, 1983, Putterman 1983, Salas-Brito et al 1990), they have not yet found a
place in many of the excellent NRQ textbooks available. In this paper we want to contribute
to alleviate the situation by first explaining how superselection rules fit in the general structure
of quantum mechanics and then by offering some examples in the context of NRQ.
2. Superselection rules in quantum theory
2.1. Superposing photons and electrons is forbidden
Let us begin by analysing the possibility of coherent superpositions of photon and electron
states. Any such superposition could, at least in principle, be written in the form
(1)
|9i = ap |photoni + ae |electroni.
Recall that whereas electrons have spin 21 , which makes them fermions, photons have spin 1,
making them bosons instead. Bosons and fermions differ in many ways, but, for our purposes,
the crucial difference is in their behaviour under rotation. Consider a bosonic state and rotate
the system by 2π, a rotation which is expected to leave the physical content of this—and any
other—state unchanged. Under this rotation the bosonic state does indeed not change; but a
fermionic state ends multiplied by −1 under that same rotation (Feynman et al 1965, Sakurai
1985). Even so, there is nothing to worry about since the quantum states are determined up to
a normalizing factor and thus the final state is also physically indistinguishable from the initial
one. However, if rather than rotating a fermionic or a bosonic state separately, we rotate the
whole superposition |9i by 2π then the rotated state becomes
R(2π)|9i = |9 0 i = ap |photoni − ae |electroni.
(2)
Given that the probabilities associated with the states (1) and (2) differ, that is, as h9|9i 6=
h9 0 |9 0 i, the superposition becomes essentially different from the original state after the
2π rotation. This implies that a superposition of the form (1) is devoid of physical meaning
unless ap = 0 or ae = 0, or, in other words, only if coherent superpositions are not allowed.
Thus, to be consistent with the properties of rotations, quantum theory is forced to restrict the
superposition principle, i.e. to impose a superposition rule between photons and electrons or,
to state the restriction in more general terms, to forbid superpositions of any fermionic with
any bosonic states whatsoever. This superselection rule is usually called univalence (Wick et
al 1952, Streater and Wightman 1964).
To avoid possible misunderstanding, it is important to indicate that there is a way of
making sense of a state like (1), not as a coherent superposition but as one of the mixed states
or incoherent superpositions. Equation (1) can have meaning as a sort of density operator (see
subsection 2.2) describing a statistical ensemble with a fractional population ap of photons
and a fractional population ae of electrons (Landau and Lifshitz 1977). For example, this can
be necessary to describe the situation within a switched-on floodlight.
2.2. Coherent versus incoherent superpositions
Before proceeding any further, it is convenient to review the difference usually made in quantum
mechanics between coherent and incoherent superpositions. Let us take as example the case of
Limitations on the superposition principle
239
spin- 21 particles; the spin of these can be found pointing up |+i or down |−i along any direction;
the most general spin state can be expressed as |φi = a+ |+i + a− |−i, where the numbers a+
and a− are assumed to comply with |a+ |2 + |a− |2 = 1. This superposition describes a state
with a well-defined spin. The value of any observable O in the pure state |φi is the quantum
expectation value
∗
∗
hφ|O|φi = |a+ |2 h+|O|+i + |a− |2 h−|O|−i + a+
a− h+|O|−i + a−
a+ h−|O|+i.
(3)
The last two terms in this equation are the interference terms; they are an unavoidable
characteristic of pure states. However, pure states are not general enough to describe every
possible situation; sometimes the need to describe a mixed state might arise. For example, if
the spin of some unfiltered beam of particles is required, a mixture is necessary to describe
the random spin orientations in the beam—note that such random behaviour has nothing to do
with quantum features.
To grasp the situation better, let us think of what we should expect of a spin measurement
in an unfiltered beam? It is clear that any measurement should produce the classical (i.e.
non-quantum) average value
hSi = + 21 h̄w+ − 21 h̄w− = w+ h+|S|+i + w− h−|S|−i
(4)
where S is the spin operator, and the numbers w+ and w− respectively stand for the ratio of
particles with spin up (+ 21 h̄) or down (− 21 h̄) in the beam. This is as we should expect since the
numbers w+ and w− are just statistical weights describing the population in the beam. This
expression may suggest that, in dealing with mixtures, it is convenient to introduce the density
operator (Landau and Lifshitz 1977)
ρ ≡ w+ |+ih+| + w− |−ih−|
(5)
to characterize the state; normalizing (5) means that w+ + w− = Tr(ρ) = 1. With such
characterization, the measurement of any property O is calculated using not the quantum
mechanical expectation value as in pure states, but by taking the trace of the product of O
times ρ:
hOi = Tr(Oρ) = w+ h+|O|+i + w− h−|O|−i.
(6)
This is the ensemble average of O in the mixture (4). The important point to note is that this
expression lacks the interference terms of the quantum expectation value (3); we can say that in
mixtures the relative phase of the individual states does not matter. As the next section shows,
in systems with superselection rules there necessarily are states which can be interpreted only
as incoherent superpositions, and, therefore, which can never interfere with one another.
2.3. Superposition rules in the formalism of quantum theory
If in a quantum system there exist a hermitian operator G which commutes with every
observable O in the system (i.e. it conmutes with the basic operators p, q, and possibly the
spin), and if G is found not to be a multiple of the identity, then, it is said that a superselection
rule induced by G operates in the system. This is in fact closely related to the failure of the
superposition principle (Streater and Wightman 1964, Lipkin 1973, Sudbery 1986) as the next
subsection shows.
Since G commutes with every observable, it commutes, in particular, with the Hamiltonian
and it necessarily corresponds to a conserved quantity. We must distinguish this property of G
from the corresponding one of ordinary conserved quantities like the energy, since physically
realizable states exist that are not eigenstates of the Hamiltonian; when a superselection rule
induced by a operator G acts on a system every physically realizable state is an eigenstate
of G and not every possible superposition can represent a physical state. The property of
commuting with every observable operator makes the variable G sharply measurable in every
situation, thus being rather similar to a c-number; known operators of this sort in NRQ are,
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for example, the mass, the electric charge, the particle number and the chirality of a molecule
(Kaempfer 1965, Lipkin 1973, Pfeifer 1980, Primas 1981).
Let us denote the eigenstates and eigenvalues of G, as |gm ; αm i and gm , respectively;
αm stands for every other quantum number needed to specify the states. As every operator
commutes with G its eigenstates can always be chosen as eigenstates of the rest of observables
of the system; thus, any state |Si can always be expressed as a superposition of the eigenstates
of G:
X
am |gm ; αm i.
(7)
|Si =
m
The Hilbert space of the system, H, is naturally decomposed into a collection of disjoint and
mutually orthogonal subspaces Hg (since they correspond to different quantum numbers of a
Hermitian operator, and thus hgs ; αs |gm ; αm i = 0), each identified by a different eigenvalue of
G. The Hg subspaces are called the coherent subspaces or sectors induced by the superselection
rule. It is now easy to see that the direct sum (Shilov 1977) of all the coherent subspaces
H = H1 ⊕ H2 ⊕ · · · ⊕ HN comprises the whole Hilbert space of the system. One must be
somewhat cautious with the statements like (7), because, although the expansions are allowed
in principle, the very existence of the superselection rule forbids any meaningful superposition
of states in different coherent subspaces. That is, a superselection rule forbids superposing
states with different values of the eigenvalue gm , as shown in subsection 2.4. An additional
consequence of the existence of a superselection rule is that not every Hermitian operator may
represent an observable. To be an observable the operator should not cause transitions between
the different coherent sectors, as we argue in subsection 2.5.
2.4. Impossibility of superposing states belonging to different coherent sectors
To show that any superposition of eigenstates of G with different eigenvalues is not
physically
P admissible (Salas-Brito 1990), consider first that, on the contrary, the state
|ui =
m um |gm ; αm i is physically sound; however, since G commutes with every other
observable, |ui should be an eigenstate of every operator in a complete set of commuting
observables of the system. Denote this set as Os , s = 1, . . . , N; thus Os |ui = os |ui. As
a consequence of the superselection rule, the state |gm ; αm i is also an eigenstate of G with
eigenvalue osm ; then, another way of express the action of Os on |ui is
X
Os |ui =
um osm |gm ; αm i.
(8)
m
However, all the |gm ; αm i being linearly independent of one another,
P the previous equations
imply that osm = os for every m value. Next consider the state |u0 i = m um exp(iδm )|gm ; αm i,
which is just |ui with the phases of its components modified, and apply the operator Os to it;
we easily obtain
X
um exp(iδm )|gm ; αm i = os |u0 i.
(9)
Os |u0 i = os
m
On comparing the previous expressions, we can see that both states, |ui and |u0 i, have precisely
the same quantum numbers; i.e. these two states are completely indistinguishable from one
another. But they are also completely different vectors in Hilbert space! The only way of
avoiding a contradiction is by altogether forbidding superposition of states in different coherent
subspaces. This proves the statement and establishes the connection between the existence
of G-operators and limitations on the superposition principle. Note also that the only way
of give content to superpositions like |ui or |u0 i is by interpreting them as mixtures, never as
pure states. Thus, in a system with superselection rules, the superposition principle only holds
within each coherent subspace; superpositions from different coherent subspaces can never
describe physical situations.
Limitations on the superposition principle
241
2.5. Non-existence of transitions between different sectors
It is very easy to show the vanishing of any matrix element of an observable O between states
in different coherent sectors Hn and Hm , m 6= n. To accomplish this, keep in mind that that
the eigestates of G can have common eigenvalues with any operator in the system, thus
hgm ; αm |O|gn ; αn i = ohgm ; αm |gn ; αn i.
(10)
However, states with different quantum numbers are orthogonal hgm ; αm |gn ; αn i = 0. Thus
no observable can ever induce transitions between states in different coherent sectors of the
Hilbert space. This means, for example, that according to the univalence superselection rule, it
is impossible to produce a single fermion from a single boson in any conceivable NRQ process.
2.6. The most general superselection rules
Superselection rules are very important properties of quantum systems. The most general
superselection rules that, it is believed, must hold in any quantum theory are associated first with
the existence of fermions and bosons, i.e. univalence, and then to every absolutely conserved
quantum number: there are superselection rules associated with the electric charge, the baryon
number, and the three lepton numbers (Lipkin 1973, Sudbery 1986). This means that all
physical states of the basic building blocks of the universe should be sharp eigenstates of the
operators associated with those properties. But even within the realm of NRQ there are very
interesting superselection rules, which can be obtained by such simple calculations that they
could be part of almost any intermediate NRQ course. This is demonstrated explicitly in the
next section.
3. Superselection rules in non-relativistic quantum mechanics
The superselection rules acting in non-relativistic quantum mechanics and the operators G
associated with them are discussed in this section. We briefly discuss first the univalence
superselection rule separating the bosonic from the fermionic sectors in the Hilbert space
of NRQ, and then the superselection rule separating the different mass sectors, which allow
the mass to be treated as just a parameter in the Schrödinger equation. The electric charge
superselection rule also operates in NRQ but, as its derivation involves field operators
(Aharonov and Susskind 1967, Strocchi and Wightman 1974), the derivation of the rule is
beyond the scope of this paper.
3.1. Univalence superselection rule
In subsection 2.2, following the arguments of Wick et al (1952), we have already sketched a
proof of univalence. It is possible to rederive this superselection rule using arguments closer
to the spirit of NRQ. This has been excellently done by Müller-Herold (1985), so here we only
want to point out a few things that do not follow directly from the discussion in subsection 2.2.
Let us mention that univalence in NRQ can be regarded as a direct consequence of the existence
of indistinguishable particles and that the operator associated with it is the permutation operator
P interchanging particles in any multiple particle state (Lipkin 1973). The whole Hilbert space
of non-relativistic quantum mechanics thus splits as the direct sum of fermionic and bosonic
coherent subspaces, i.e. as the direct sum of the eigenspaces of P .
3.2. Superselection rule separating different mass values
In dealing with the Schrödinger equation, mass is mostly assumed to be a c-number (mass is
sometimes assumed to be an operator, for example, it is convenient to do so for describing
the propagation of electrons in a crystal (Ashcroft and Mermin 1976)); moreover, states
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superposing different masses are conspicuously absent in NRQ. Mass seems to comply with a
superselection rule; this follows from the Galilean invariance of NRQ as originally noted by
Bargman (1954).
The change in a state function when both a displacement of the origin by the constant
quantity r and a change between two reference frames moving with respect to one another
at a constant velocity v occur simultaneously (that is, when the Galilean transformation q 0 =
q +v t+r is performed) can be represented by the unitary operator UG = exp(imv·q 0 −imv 2 t/2)
(Kaempfer 1965, Landau and Lifshitz 1977). The action of UG on a quantum state |9i is
(11)
|9 0 i = UG |9i = exp im(v · q 0 − v 2 /2) |9i.
Therefore h9 0 |9 0 i = h9|9i and Galilean invariance is manifest in non-relativistic quantum
mechanics. Selecting r = 0, specializes (11) to the operator associated with a pure Galilean
transformation, Uv ; whereas selecting v = 0, specializes it to the operator associated with a
pure spatial displacement, Ur . It must be clear that the combined action on |9i of a spatial
displacement Ur , a pure Galilean transformation Uv , the opposite displacement U−r and the
inverse Galilean transformation U−v , just amounts to an identity transformation. Its net effect
is again just a change in the phase of the state: UI |9i = exp(imv · r )|9i, where we have
defined UI ≡ U−v U−r Uv Ur . The transformation UI is seen to be of trivial consequences
as long as we are dealing with definite mass states; the change becomes crucial, however, if
we consider states |φ1 i and |φ2 i corresponding to different masses m1 and m2 . Since any
superposition |φi = a1 |φ1 i + a2 |φ2 i transforms under the identity-equivalent transformation
UI as
|φ 0 i ≡ UI |φi = a1 exp(im1 r · v )|m1 i + a2 exp(im2 r · v )|m2 i
(12)
0
you can see that if m1 6= m2 the states |φi and |φ i become essentially different. We have to
conclude that mass induces a superselection rule in NRQ and, therefore, that mass can always
be sharply measured in any situation. Thus NRQ is utterly inadequate for describing states
endowed with a mass spectrum or for describing transformations between particles.
3.3. Particle number superselection rule
Given the mass superselection rule, it is very easy to obtain the closely related superselection
rule separating states with different particle numbers in Fock space (Lipkin 1973). The proof
follows from the following argument (Müller-Herold 1985), given a state |ni describing
n identical particles, each with mass m, this state should transform, under the Galilean
transformation U (I ) of subsection 2.2, as U (I )|ni = exp(inmv · r )|ni. From the fact that
the states of n particles have a total mass nm different, in general, from the mass n0 m of n0 particle states, the argument of the previous section unavoidably leads to the conclusion that
any superposition of n-particle with n0 -particle (n 6= n0 ) states must be a mixed state. The
G-operator associated with this superselection rule is the particle number operator N .
4. Summary
Superselection rules have been introduced and its principal consequences have been discussed
within the framework of non-relativistic quantum mechanics. We have exhibited how certain
quantum requirements are capable of constraining the applicability of the superposition
principle. Using the basic machinery of quantum mechanics we have derived three of the
basic superselection rules that operate in that theory, namely, univalence, mass and particle
number. It is surprising that these very interesting and important ideas are almost never
discussed in an intermediate course in quantum mechanics. Our explanation for this curious
situation is that, originally, superselection rules were introduced in the context of quantum
field theory and they were considered features of infinite dimensional theories. On the other
hand, perhaps some theoretical prejudices made people believe that it was impossible to find
Limitations on the superposition principle
243
superselection rules in simple quantum systems. However, we now know that one can find
them even in rather simple systems; for example, in the one-dimensional hydrogen atom with
interaction V (x) = −e2 /|x| where e is the electronic charge (Núñez-Yépez et al 1988, 1989,
1987, Martı́nez-y-Romero et al 1989).
Acknowledgments
This work has been partially supported by CONACyT (grant 1343P-E9607). ALSB wants to
thank J L Carrillo-Estrada and L Fuchs-Gómez for their one-week hospitality at, respectively,
the Instituto de Fı́sica and the Facultad de Ciencias Fı́sicas y Matemáticas of the BUAP where
this work was partially written. We are also extremely grateful to P Pfeifer for providing us
with a copy of his dissertation. The useful comments of P A Terek, G Billi, K Quiti and
F C Bonito are gratefully acknowledged. For their help in revising the typescript we must
thank U Sasi, F Sadi, N Kuro and B Caro.
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